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Convex optimization for the synthesis of matching filters
Gibin Bose, Fabien Ferrero, Leonardo Lizzi, Fabien Seyfert, David Martínez
Martínez
To cite this version:
Gibin Bose, Fabien Ferrero, Leonardo Lizzi, Fabien Seyfert, David Martínez Martínez.
Con-vex optimization for the synthesis of matching filters.
ICEAA 2017- International
Confer-ence on Electromagnetics in Advanced Applications, Sep 2017, Verona, Italy.
pp.1450-1453,
�10.1109/ICEAA.2017.8065554�. �hal-01621215�
Convex Optimization for the Synthesis of Matching
Filters
G. Bose
∗F. Ferrero
†L. Lizzi
†D. Martinez
∗F. Seyfert
∗Abstract — In this work we study a particular fil-ter synthesis problem in order to minimize the re-flection coefficient of the global system consisting of filter and antenna. The matching problem is formu-lated as an optimization problem involving the min-imization of a pseudo hyperbolic distance and the solution to this problem using H∞approach yields a lower bound for the matching criterion related to the computation of a matching filter, with prescribed fi-nite degree, under selectivity constraints.
1 INTRODUCTION
In classical communication systems, antennas used in the reception or emission chain are often associ-ated with a matching network followed by a band-pass filter so as to select a proper frequency chan-nel, to reject unwanted signals and to maximize the power transfer. In the last years, to fulfill the con-straints imposed by modern applications (e.g., the internet-of-things (IoT), wireless sensor networks, e-health, etc.), wireless communicating devices are required to be compact and energy autonomous to enable easy integration and long-lasting operation. In order to improve the energy efficiency as well as the footprint of the transmission chain we pro-pose to design, in a single circuit, the filter and the matching network. This matching filter has to be design accordingly to the antenna. This co-design approach amounts to tackle a particular fil-ter synthesis problem where one of the filfil-ter’s port is loaded by a frequency varying load, namely the antenna (Fig. 1).
The matching problem, is a rather old one, and goes back to the foundational work of Fano and Youla in the fifties [1]. This theory develops a syn-thesis procedure for matching networks, based on the derivation of a 2 port loss-less scattering ma-trix representing the filter chained to the antenna. However unlike in the Darlington’s insertion-loss synthesis, this framework does not translate into a convex or quasi-convex optimization problem of Zolotariov type yielding an optimal response. This is mainly due to the interpolation constraints that
∗Inria Sophia Antipolis Mediterranean, 2004 Route des Lucioles - BP 93 06902 Sophia Antipolis Cedex, France, e-mail: gibin.bose@inria.fr, tel.: +33 4 92 38 77 77.
†Universit´e Cˆote d’Azur, CNRS, LEAT, Campus Sophi-aTech Bˆatiment Forum Haut, 930 Route des Colles, 06410 Biot , France, e-mail: fabien.ferrero@unice.fr, tel.: +33 4 92 94 28 04, fax: +33.(0)4.92.94.28.99. L F S22 L11 S11 L22 G11 G22
Figure 1: Matching Filter (S) and Load (L)
needs to be added to the two port’s description to take into account the presence of the antenna. For this reason, this theory was progressively aban-doned to the benefit of a non-convex optimiza-tion method called real frequency technique and originally introduced by Carlin [2]. Although this heuristical approach seems to give reasonable re-sults in practice, no result is known about the global optimality or near-optimality of the so obtained matching network. More recently J.W Helton pro-posed an H∞approach to the problem where an in-finite dimensional matching network is being sought for [3]. The matching problem is recast as a quasi-convex optimization problem involving the mini-mization of a pseudo hyperbolic distance. The absence of any degree constraint on the circuital response is here traded for the guaranteed global optimality of the obtained response. The relative mathematical complexity of this procedure coupled to the impossibility to realize in practice an infinite dimensional H∞response have severely limited its impact in electronics.
In this work we study a practical implementa-tion of J.W Helton’s approach based on the reso-lution of a bounded extremal problem in H∞. De-tails are in particular given about the effective com-putation of the best approximation operator from L∞→ H∞ via Nehari’s theory. We show that the
solution to this problem furnishes a lower bound for the matching criterion related to the computation of a matching filter, with prescribed finite degree, under selectivity constraints. When the antenna’s reflection parameter admits a rational approxima-tion of degree one in the frequency band of interest, we have recently shown that the best matching fil-ter synthesis problem admits a convex formulation. This approximation well applies to miniature and narrowband antennas suitable for compact IoT de-vices requiring the transmission of a limited amount of data. We studied on concrete antenna examples
the computation and behaviour of the best finite dimensional matching filter response using Helton’s approach.
2 MATCHING PROBLEM
We consider the synthesis of a matching filter for a given frequency varying load. Given a passband B the matching filter is designed so as to minimize, when plugged on the load, the power reflected by the load. The global system (G) consisting of fil-ter together with the load is represented in Fig. 1. The parameters G11, S11and L11denote the input
reflection coefficient of the global system, match-ing filter and load respectively whereas G22, S22
and L22 represent the output reflection coefficient
of the same respectively.
If the filter is considered lossless, the output re-flection coefficient of the global system (at each fre-quency ω) can be computed as [5] :
G22= L22+ L12L21S22 1 − S22L11 = L22− S22det(L) 1 − S22L11 = det(L) L ∗ 11− S22 1 − S22L11 (1) Hence the modulus of the output reflection coeffi-cient of the global system is obtained as the pseudo hyperbolic distance between S22 and L∗11 :
|G22| = S22− L∗11 1 − S22L11 = δ (S22, L∗11) (2) 2.1 Optimization Problem
In the infinite dimensional setting, where the filter’s scattering parameters are sought for in the Hardy space H∞ of bounded analytic functions of the lower half plane [6], the matching problem over a frequency band B can be formulated as follows.
Problem 1 : Minimize the pseudo hyperbolic distance between S22 and L∗11.
min max
ω∈Bδ(S22(ω), L ∗
11(ω)) (3)
subject to: S22(ω) ∈ H∞ and |S22(ω)|ω∈R≤ 1
(4) where B is the desired matching band (passband). 2.1.1 Practical Algorithm
We suppose that we possess a passive rational model f of the antenna’s reflection parameter L11,
obtained for example via rational approximation
−1 0 1 2 −20 −15 −10 −5 0 Normalized Frequency Reflection P arameter (dB) Ref 1 Ref 2 Ref 3 Ref 4
Figure 2: Reference Functions
techniques at hand of scattering measurements. We form a family of rational reference functions {kα}
parametrized by α ∈ R+, the modulus of which
mimic an ideal step function kopt:
kopt(ω) =
L ω ∈ B 0 ≤ L ≤ 1
1 ω /∈ B (5)
There are multiple ways to approach rationally a step function. We choose here to follow the clas-sical Darlington insertion loss synthesis for filters. Consider the Belevitch form of a general loss-less rational scattering matrix,
G = 1 q p∗ −r∗ r p (6) where is a unimodular constant, q satisfies the spectral equation qq∗ = pp∗+ rr∗. The modulus square of G22can be expressed as :
|G22|2= pp∗ qq∗ (7) = pp ∗ pp∗+ rr∗ = 1 1 +rrpp∗∗ (8)
For simplicity, we will consider here the case where r is of degree zero. We fix α = rr∗, and define kα
to be the rational outer factor verifying: |kα|2=
1
1 + α|1p|2 (9)
If we take for p, p = TN the Tchebyschev
polyno-mial of degree N in the interval B, {kα} forms a
family of Tchebyschev rational reference functions whose modulus can be varied monotonously using the parameter α.
Alternatively, if we take p = TN + c, where c
is a positive constant the {kα}0s form a family
of rational functions with monotonously decreasing moduli, that equi-oscillate on B between the values 1/(1 + α/(c + 1)2) and 1/(1 + α/(c)2). Fig. 2 shows
some examples of reference functions where Ref 1 and Ref 2 are equi-oscillating references whereas Ref 3 and Ref 4 are pure Tchebyschev references. 2.1.2 Bound for the reflection level using
H∞ approach
After deriving the rational function f and forming the family of rational reference functions {kα}, we
approach Problem 1 as follows :
Problem 2 : Find a g ∈ H∞ such that:
sup
ω∈R
δ(f∗(ω), g(ω)) ≤ |k(ω)| (10) where k runs over the family {kα}.
Solution : The hyperbolic disk, δ(f∗, g) ≤ |k|
with centre f∗ and radius |k| translates to the fol-lowing euclidean disk :
g − (1 − |k| 2) 1 − |k|2|f |2f ∗ ≤ (1 − |f | 2) 1 − |k|2|f |2|k| (11)
Now, if we factorize 1−|k|(1−|f |2|f |2)2 = uu∗, where u is
an outer function and consider the outer function v = u2k, |v| = (1 − |f |2) 1 − |k|2|f |2 |k| (12) Now dividing (11) by |v|, g v − 1 v (1 − |k|2) 1 − |k|2|f |2f ∗ ≤ 1 (13)
yields a classical Nehari problem. Let G = gv and F = v11−|k|(1−|k|2|f |2)2f∗.
The function v being invertible in H∞, the prob-lem reduces to
min ||G − F ||∞ (14)
subject to : G ∈ H∞ and finding the G at which the infimum is attained. The solution to (14) can be obtained using the classical operator theoretic approach of Nehari [6],
G = F −HF(V )
V (15)
where HF is the Hankel operator with symbol F
and V one of its maximizing vectors.
For numerical reasons we chose to implement Nehari’s solution to the extremal problem (14) in
the framework of Hardy spaces of the unit disc D. This is done classically using the conformal map z → j(z − 1)/(z + 1) sending the unit disk to the lower half plane. Given a rational function F , in order to find the maximizing vector of the Han-kel operator, HF : H2(D) → ¯H2(D), we follow the
steps below:
(i) Let {zj} be the poles of F inside the unit disk.
(ii) gj = {1−z ¯1zj} form the basis of (Ker(HF))⊥
and hj = {z−z1
j} form a basis of the image of
HF in ¯H2.
(iii) Let the gram matrix of the {gj}0s be,
G1= [ai,j] =< gi, gj >=
1 1 − ¯zjzi
(16) and the gram matrix of the {hj}0s,
G2= [bi,j] =< hi, hj>=
1 1 − ¯zizj
(17)
(iv) Denote the matrix of the Hankel operator by A in the basis {gj} and {hj}, and solve the
gener-alized eigenvalue problem: A∗G2Au = λG1u.
The eigenvector corresponding to the largest eigenvalue will provide the maximizing vector of the Hankel operator and the square root of largest eigenvalue will provide the value of the minimum in (14). The function g is obtained multiplying back by the outer factor v, that is g = vG.
So for a given f , the solution to Problem 2, when one exists, provides the output reflection coefficient S22of a matching filter for a given reference kα. As
shown by equation (13), there exists a solution to Problem 2, if and only if the operator norm of HF
is less-than or equal to unity. Increasing the param-eter α lowers the level of reference, implying that ||HF|| is an increasing function of α (when k = kα
is set). Ruling out the case where f is a constant, and noting that an anti-analytic function can not be approached uniformly and arbitrarily close by an analytic one, implies that there exists α0 such
that ||Hf|| = 1. For α = α0, equality holds in
equa-tion (10). We call the so obtained reflecequa-tion coef-ficient, an approximate solution to Problem 1 with respect to the family of rational references {kα}. A
remarkable property of the latter is that its degree is comparable to that of the reference kα0 (proof of
this goes beyond the objective of this paper). 3 Results
For α = α0, the modulus of the global system’s
780 800 820 840 860 880 −8 −6 −4 −2 0 Frequency (MHz) Reflection P arameter (dB) G22 L11 Reference Level (-5.85)
Figure 3: Optimal System reflection with degree 9 equi-oscillating reference for an antenna of degree 1 with Fano Bound = -6.20.
780 800 820 840 860 880 −15 −10 −5 0 Frequency (MHz) Relection P arameter (dB) G22 L11 Reference Level (-7.73)
Figure 4: Optimal System reflection with degree 6 Tchebyschev reference for an antenna of degree 2 with Fano Bound = -13.64.
the optimal overall system reflection for an antenna of degree one using equi-oscillating references of de-gree nine, while Fig. 4 shows the same for an an-tenna of degree two using Tchebyschev references of degree six. Optimal system reflection level obtained using equi-oscillating references of various degrees were computed for a given antenna of degree one (Fig. 5).
4 Conclusion
Helton’s H∞ approach to matching theory, sup-posed to yield a guaranteed optimal response at the cost of an infinite degree matching network, can be successfully adapted to yield finite degree matching networks by using families of finite degree
ratio-1 2 3 4 5 6 7 8 −6 −5 −4 Filter Degree Reflection Lev el (dB)
Figure 5: Optimal System reflection for different degrees of filters using equioscillating references (Degree 1 antenna with Fano Bound = -6.20).
nal references. The procedure can therefore be put into practice to derive matching networks for mis-matched antennas. Whether the one dimensional families of reference functions can be extended to broader functional manifolds, while preserving the underlying convexity of the optimization problem to solve is currently under study.
Acknowledgments
This work was supported by the LABEX1
UCN@Sophia supervised by the French Ministry for Research and Higher Education.
References
[1] R.M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances”, Technical Report Volume 249, MIT, 1950. [2] H.J. Carlin and P.P. Civalleri, “Wideband
Cir-cuit Design”, CRC Press, 1997.
[3] J. William Helton, “Non-euclidean functional analysis and electronics”, Bull. Amer. Math. Soc. (N.S.), 7(1):1-64, 07 1982.
[4] J. William Helton, “Broadbanding: Gain equal-ization Directly From Data”, IEEE Transac-tions on circuits and systems, 1981.
[5] L. Baratchart, M. Olivi, and F. Seyfert, “Boundary Nevanlinna–Pick Interpolation with Prescribed Peak Points. Application to Impedance Matching ”, SIAM Journal on Mathematical Analysis, 2017, Vol. 49, pp. 1131-1165
[6] John B. Garnett, “Bounded Analytic Func-tions”, Springer, 2007.
